# Meaning of vector calc identity

I want to prove that $$(\textbf{a}\times\nabla)\cdot\textbf{b}=-\nabla\cdot(\textbf{a}\times\textbf{b})+\textbf{b}\cdot(\nabla\times\textbf{a})$$ using the levi civita symbol, but what does $$(\textbf{a}\times\nabla)\cdot\textbf{b}$$ even mean in general? If $$\textbf{a}=(a_1,a_2,a_3)$$ is $$(\textbf{a}\times\nabla)_1=a_2\partial_3-a_3\partial_2$$ (and so on for other components). When you dot this with $$\textbf{b} = (b_1,b_2,b_3)$$ do you get (for component 1) $$(a_2\frac{\partial b_1}{\partial x_3}-a_3\frac{\partial b_1}{\partial x_2})$$?

• Quite impressive mathjax for a "new user"! Commented Sep 26, 2018 at 17:34
• How can you prove the identity, when you don't even know what $(\textbf{a}\times\nabla)\cdot\textbf{b}$ even means? Commented Sep 26, 2018 at 17:35
• @amWhy I think 11Elves asked the meaning of the formula (why would you ever use such thing). Is there maybe a physical significance of such an expression? He/she knows to apply the steps to prove it, but it would be just a mechanical process, without any meaning. Commented Sep 26, 2018 at 17:40
• I want to prove it, but first i need to know what the left side means. Its not a curl, but something strange. Help Commented Sep 26, 2018 at 18:43
• @Andrei At least, that identity is involved in "Energy Conservation" in "Classical Elegtromagnetism". However, I guess there's something wrong with your post. See this link. Commented Sep 26, 2018 at 19:13

$$\mathbf{a}\times \nabla$$ is indeed the operator $$(a_2 \partial_3-a_3\partial_2, a_3\partial_1 -a_1\partial_3, a_1\partial_2-a_3\partial_2)$$. Taking the dot product with $$\mathbf{b}$$ gives $$(\mathbf{a}\times \nabla)\cdot \mathbf{b}=a_2\partial_3 b_1-a_3\partial_2 b_1 + a_3\partial_1 b_2-a_1\partial_3 b_2 + a_1\partial_2 b_3 - a_2\partial_1 b_3$$. However, we can also write this as $$a_1(\partial_2 b_3-\partial_3 b_2)+a_2(\partial_3b_1-\partial_1b_3)+a_3(\partial_1 b_2-\partial_2 b_1)=\mathbf{a}\cdot (\nabla \times \mathbf{b})$$, which leads you to the formula mentioned by Felix Marin above.